A Spectral General Circulation Model Using a Piecewise-Constant Finite-Element Representation on a Hybrid Vertical Coordinate System

Abstract

A numerical scheme for the vertical discretization of primitive equations in a generalized pressure-type coordinate is developed through application of the Galerkin formalism with piecewise-constant finite elements: this methodology affords an elegant—and direct—mean of formulating conservative discretization schemes without the arbitrariness that usually characterizes the development of finite differences. The form of the resulting semidiscrete equations is equivalent to some second-order accurate finite-difference approximation to the continuous equations. Flexibility of this scheme in the choice of different layers for projecting the thermodynamic and momentum variables effectively allows for staggering of these variables in the vertical. Numerical integrations performed with this scheme at various vertical resolutions have revealed the sensitivity of the simulated circulation to resolution in the lower stratosphere. We found that application of the “lid” upper boundary condition at a finite height alleviates a documented bias in the estimation by this scheme of the thermal wind relationship at upper level with coarse resolution, and this is accomplished here without sacrificing the conservation properties of the scheme.